I really need help solving the following question:
Given: $$f(n) = o(g(n))$$
Prove: $$3^{f(n)} = o(3^{g(n)})$$
My attempt:
I know that $\frac{f(n)}{g(n)} \xrightarrow{} 0 $.
I need to prove that $f(n) - g(n) \xrightarrow{} -\infty$ so that $3^{f(n)-g(n)} \xrightarrow{} 0$.
How do I prove that?
In general, I'm not sure what properties can I assume about $f$ and $g$. Are they positive? What else do I know about them? Are we assuming that all these function approach $\infty$ or can they be $\frac{1}{n}$ etc.?